Subgroups — C₂²×D₃×#634 #13181589

Order 1728.

5724 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6308	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×C₄×D₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6293	C₂	1	2	yes	maximal, complemented
5	C₂²×D₃×C₃⋊Dic₃	C₂	1	2	yes	maximal
6	C₂×D₃×#634	C₂	1	24	yes	maximal, direct product factor
7	C₂³×#634	—	3	1	no	maximal
8	C₂²×D₃×Dic₆	—	3	4	no	maximal
9	C₂³×C₃²×D₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×#6308	C₂²	1	12	yes	complemented
13	C₂²×C₃×C₃⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×#634	C₂²	1	12	yes	complemented
15	C₂²×C₃²×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃²×C₄×D₃	C₂²	1	12	yes	—
17	C₂×#6293	C₂²	1	24	yes	complemented
18	C₂×D₃×C₃⋊Dic₃	C₂²	1	24	yes	—
19	D₃×#634	C₂²	1	64	yes	direct product factor
20	C₂³×C₃²×C₄	—	3	1	no	—
21	C₂³×C₃⋊Dic₃	—	3	2	no	—
22	C₂²×#634	—	3	1	no	—
23	C₂²×#634	—	3	2	no	—
24	C₂²×#634	—	3	24	no	—
25	C₂²×#634	—	3	4	no	—
26	C₂²×#634	—	6	4	no	—
27	C₂²×#634	D₃	1	1	yes	characteristic, direct product factor
28	C₂²×C₃×C₄×D₃	D₃	1	4	yes	—
29	C₂²×C₃×Dic₆	—	3	4	no	—
30	C₂²×#627	—	3	8	no	—
31	C₂²×D₃×Dic₃	—	3	8	no	—
32	C₂×D₃×Dic₆	—	3	96	no	—
33	C₂³×C₃³	C₂³	1	1	yes	characteristic
34	C₂²×C₃²×D₃	Q₈	1	8	yes	complemented
35	C₂²×C₃²×D₃	C₂³	1	6	yes	—
36	C₂×C₃³×C₄	C₂³	1	6	yes	—
37	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	12	yes	—
38	#6308	C₂³	1	16	yes	complemented
39	C₂×C₃×C₃⋊Dic₃	C₂³	1	12	yes	—
40	C₃×#634	C₂³	1	16	yes	complemented
41	C₂×C₃²×Dic₃	C₂³	1	6	yes	—
42	C₃²×C₄×D₃	C₂³	1	16	yes	—
43	#6293	C₂³	1	32	yes	complemented
44	D₃×C₃⋊Dic₃	C₂³	1	32	yes	—
45	C₂²×D₃×Q₈	—	9	1	no	—
46	C₂³×Dic₆	—	9	4	no	—
47	C₂⁴×C₃²	—	3	1	no	—
48	C₂³×C₃×D₃	C₂×D₃	1	4	yes	—
49	C₂²×C₃²×C₄	—	2	4	no	—
50	C₂²×C₃²×C₄	—	3	1	no	—
51	C₂²×C₃²×C₄	—	3	12	no	—
52	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
53	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃⋊Dic₃	—	3	2	no	—
55	C₂²×C₃⋊Dic₃	—	3	8	no	—
56	C₂²×C₃⋊Dic₃	—	3	24	no	—
57	C₂²×C₃⋊Dic₃	—	6	8	no	—
58	C₂²×C₃⋊Dic₃	C₂×D₃	1	2	yes	—
59	C₂×#634	—	3	12	no	—
60	C₂×#634	—	3	48	no	—
61	C₂×#634	—	3	24	no	—
62	C₂×#634	—	3	64	no	—
63	C₂×#634	—	6	48	no	—
64	C₂×#634	C₂×D₃	1	12	yes	direct product factor
65	C₂²×C₃×Dic₃	—	3	8	no	—
66	C₂²×C₃×Dic₃	C₂×D₃	1	4	yes	—
67	C₂×C₃×C₄×D₃	C₂×D₃	1	48	yes	—
68	C₂×C₃×Dic₆	—	3	48	no	—
69	C₂×#627	—	3	96	no	—
70	C₂×D₃×Dic₃	—	3	96	no	—
71	D₃×Dic₆	—	3	256	no	—
72	C₂²×C₃³	C₂×Q₈	1	4	yes	complemented
73	C₂²×C₃³	C₂⁴	1	3	yes	—
74	C₂×C₃²×D₃	C₂×Q₈	1	24	yes	complemented
75	C₂×C₃²×D₃	C₂⁴	1	4	yes	—
76	C₃³×C₄	C₂⁴	1	4	yes	—
77	C₃⋊(C₃⋊Dic₃)	C₂⁴	1	8	yes	—
78	C₃×C₃⋊Dic₃	C₂⁴	1	8	yes	—
79	C₃²×Dic₃	C₂⁴	1	4	yes	—
80	C₂²×C₃×Q₈	—	9	1	no	—
81	C₂³×C₃×C₄	—	3	4	no	—
82	C₂³×Dic₃	—	9	8	no	—
83	C₂×D₃×Q₈	—	9	24	no	—
84	C₂²×C₄×D₃	—	9	2	no	—
85	C₂²×C₄×D₃	C₃⋊D₃	1	1	yes	characteristic
86	C₂²×Dic₆	—	3	4	no	—
87	C₂²×Dic₆	—	9	96	no	—
88	C₂²×Dic₆	—	9	8	no	—
89	C₂²×Dic₆	—	9	4	no	—
90	C₂²×Dic₆	—	9	1	no	—
91	C₂²×Dic₆	—	9	2	no	—
92	C₂²×Dic₆	—	18	4	no	—
93	C₂³×C₃²	—	2	4	no	—
94	C₂³×C₃²	—	3	8	no	—
95	C₂³×C₃²	—	3	6	no	—
96	C₂³×C₃²	C₂²×D₃	1	4	yes	—
97	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×D₃	Dic₆	1	32	yes	complemented
99	C₂²×C₃×D₃	C₂²×D₃	1	24	yes	—
100	C₂×C₃²×C₄	—	2	24	no	—
101	C₂×C₃²×C₄	—	3	6	no	—
102	C₂×C₃²×C₄	—	3	16	no	—
103	C₂×C₃²×C₄	C₂²×D₃	1	24	yes	—
104	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
105	C₂×C₃⋊Dic₃	—	3	12	no	—
106	C₂×C₃⋊Dic₃	—	3	32	no	—
107	C₂×C₃⋊Dic₃	—	3	48	no	—
108	C₂×C₃⋊Dic₃	—	6	48	no	—
109	C₂×C₃⋊Dic₃	C₂²×D₃	1	12	yes	—
110	#634	—	3	16	no	—
111	#634	—	3	32	no	—
112	#634	—	3	64	no	—
113	#634	—	6	64	no	—
114	#634	C₂²×D₃	1	16	yes	direct product factor
115	C₂×C₃×Dic₃	—	3	48	no	—
116	C₂×C₃×Dic₃	C₂²×D₃	1	24	yes	—
117	C₃×C₄×D₃	C₂²×D₃	1	64	yes	—
118	C₃×Dic₆	—	3	64	no	—
119	#627	—	3	128	no	—
120	D₃×Dic₃	—	3	128	no	—
121	C₂³×Q₈	—	27	1	no	Sylow
122	C₂×C₃³	C₂²×Q₈	1	6	yes	complemented
123	C₂×C₃³	C₂⁵	1	1	yes	characteristic
124	C₃²×D₃	C₂²×Q₈	1	8	yes	complemented
125	C₂⁴×C₃	—	3	4	no	—
126	C₂³×D₃	C₂×C₃⋊D₃	1	1	yes	characteristic
127	C₂×C₃×Q₈	—	9	12	no	—
128	C₂²×C₃×C₄	—	2	4	no	—
129	C₂²×C₃×C₄	—	3	4	no	—
130	C₂²×C₃×C₄	—	3	48	no	—
131	C₂²×C₃×C₄	—	9	2	no	—
132	C₂²×C₃×C₄	D₃²	1	4	yes	—
133	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
134	C₂²×Dic₃	—	3	8	no	—
135	C₂²×Dic₃	—	9	96	no	—
136	C₂²×Dic₃	—	9	8	no	—
137	C₂²×Dic₃	—	9	2	no	—
138	C₂²×Dic₃	—	18	8	no	—
139	C₂²×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
140	D₃×Q₈	—	9	64	no	—
141	C₂×C₄×D₃	—	9	24	no	—
142	C₂×C₄×D₃	C₂×C₃⋊D₃	1	12	yes	—
143	C₂×Dic₆	—	3	48	no	—
144	C₂×Dic₆	—	9	256	no	—
145	C₂×Dic₆	—	9	96	no	—
146	C₂×Dic₆	—	9	48	no	—
147	C₂×Dic₆	—	9	12	no	—
148	C₂×Dic₆	—	9	24	no	—
149	C₂×Dic₆	—	18	48	no	—
150	C₂²×C₃²	—	2	12	no	—
151	C₂²×C₃²	—	2	16	no	—
152	C₂²×C₃²	—	3	24	no	—
153	C₂²×C₃²	—	3	4	no	—
154	C₂²×C₃²	C₂×Dic₆	1	16	yes	complemented
155	C₂²×C₃²	D₃×Q₈	1	4	yes	complemented
156	C₂²×C₃²	C₂³×D₃	1	12	yes	—
157	C₂²×C₃²	C₂³×D₃	1	3	yes	—
158	C₂×C₃×D₃	C₂×Dic₆	1	96	yes	complemented
159	C₂×C₃×D₃	C₂³×D₃	1	16	yes	—
160	C₃²×C₄	—	2	16	no	—
161	C₃²×C₄	—	3	4	no	—
162	C₃²×C₄	C₂³×D₃	1	16	yes	—
163	C₃²×C₄	C₂³×D₃	1	4	yes	—
164	C₃⋊Dic₃	—	3	8	no	—
165	C₃⋊Dic₃	—	3	32	no	—
166	C₃⋊Dic₃	—	6	32	no	—
167	C₃⋊Dic₃	C₂³×D₃	1	8	yes	—
168	C₃×Dic₃	—	3	32	no	—
169	C₃×Dic₃	C₂³×D₃	1	16	yes	—
170	C₂²×Q₈	—	9	1	no	—
171	C₂²×Q₈	—	27	2	no	—
172	C₂²×Q₈	—	27	24	no	—
173	C₂²×Q₈	—	27	1	no	—
174	C₂³×C₄	—	3	1	no	—
175	C₂³×C₄	—	27	2	no	—
176	C₃³	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₃	—	2	4	no	—
178	C₂³×C₃	—	3	24	no	—
179	C₂³×C₃	—	3	32	no	—
180	C₂³×C₃	C₂×D₃²	1	4	yes	—
181	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
182	C₂²×D₃	#634	1	8	yes	direct product factor
183	C₂²×D₃	C₂²×C₃⋊D₃	1	6	yes	—
184	C₃×Q₈	—	9	16	no	—
185	C₂×C₃×C₄	—	2	24	no	—
186	C₂×C₃×C₄	—	3	24	no	—
187	C₂×C₃×C₄	—	3	64	no	—
188	C₂×C₃×C₄	—	9	12	no	—
189	C₂×C₃×C₄	C₂×D₃²	1	24	yes	—
190	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	6	yes	—
191	C₂×Dic₃	—	3	48	no	—
192	C₂×Dic₃	—	9	128	no	—
193	C₂×Dic₃	—	9	48	no	—
194	C₂×Dic₃	—	9	12	no	—
195	C₂×Dic₃	—	18	48	no	—
196	C₂×Dic₃	C₂²×C₃⋊D₃	1	6	yes	—
197	C₄×D₃	—	9	32	no	—
198	C₄×D₃	C₂²×C₃⋊D₃	1	16	yes	—
199	Dic₆	—	3	64	no	—
200	Dic₆	—	9	128	no	—
201	Dic₆	—	9	64	no	—
202	Dic₆	—	9	16	no	—
203	Dic₆	—	9	32	no	—
204	Dic₆	—	18	64	no	—
205	C₂×C₃²	—	2	24	no	—
206	C₂×C₃²	—	2	4	no	—
207	C₂×C₃²	—	3	8	no	—
208	C₂×C₃²	C₂²×Dic₆	1	24	yes	complemented
209	C₂×C₃²	C₂×D₃×Q₈	1	6	yes	complemented
210	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
211	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
212	C₃×D₃	C₂²×Dic₆	1	32	yes	complemented
213	C₂⁴	—	3	1	no	—
214	C₂×Q₈	—	9	12	no	—
215	C₂×Q₈	—	27	24	no	—
216	C₂×Q₈	—	27	64	no	—
217	C₂×Q₈	—	27	12	no	—
218	C₂²×C₄	—	3	1	no	—
219	C₂²×C₄	—	3	12	no	—
220	C₂²×C₄	—	9	2	no	—
221	C₂²×C₄	—	27	24	no	—
222	C₂²×C₄	—	27	2	no	—
223	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
224	C₂²×C₃	—	2	12	no	—
225	C₂²×C₃	—	2	16	no	—
226	C₂²×C₃	—	3	96	no	—
227	C₂²×C₃	—	3	16	no	—
228	C₂²×C₃	D₃×Dic₆	1	16	yes	complemented
229	C₂²×C₃	C₂×#634	1	4	yes	complemented
230	C₂²×C₃	C₂²×D₃²	1	12	yes	—
231	C₂²×C₃	C₂³×C₃⋊D₃	1	3	yes	—
232	C₂×D₃	C₂×#634	1	24	yes	direct product factor
233	C₂×D₃	C₂³×C₃⋊D₃	1	4	yes	—
234	C₃×C₄	—	2	16	no	—
235	C₃×C₄	—	3	16	no	—
236	C₃×C₄	—	9	8	no	—
237	C₃×C₄	C₂²×D₃²	1	16	yes	—
238	C₃×C₄	C₂³×C₃⋊D₃	1	4	yes	—
239	Dic₃	—	3	32	no	—
240	Dic₃	—	9	32	no	—
241	Dic₃	—	9	8	no	—
242	Dic₃	—	18	32	no	—
243	Dic₃	C₂³×C₃⋊D₃	1	4	yes	—
244	C₃²	—	2	4	no	—
245	C₃²	C₂³×Dic₆	1	4	yes	complemented
246	C₃²	C₂²×D₃×Q₈	1	1	yes	characteristic, complemented
247	C₂³	—	3	6	no	—
248	C₂³	—	3	8	no	—
249	C₂³	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
250	Q₈	—	9	16	no	—
251	Q₈	—	27	32	no	—
252	Q₈	—	27	16	no	—
253	C₂×C₄	—	3	6	no	—
254	C₂×C₄	—	3	16	no	—
255	C₂×C₄	—	9	12	no	—
256	C₂×C₄	—	27	32	no	—
257	C₂×C₄	—	27	12	no	—
258	C₂×C₄	C₂×D₃×C₃⋊D₃	1	6	yes	—
259	C₂×C₃	—	2	24	no	—
260	C₂×C₃	—	2	4	no	—
261	C₂×C₃	—	3	32	no	—
262	C₂×C₃	C₂×D₃×Dic₆	1	24	yes	complemented
263	C₂×C₃	C₂²×#634	1	6	yes	complemented
264	C₂×C₃	C₂³×D₃²	1	4	yes	—
265	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
266	D₃	C₂²×#634	1	8	yes	direct product factor
267	C₂²	—	3	24	no	—
268	C₂²	—	3	4	no	—
269	C₂²	D₃×#634	1	4	yes	direct product factor
270	C₂²	C₂²×D₃×C₃⋊D₃	1	3	yes	—
271	C₄	—	3	4	no	—
272	C₄	—	9	8	no	—
273	C₄	—	27	8	no	—
274	C₄	C₂²×D₃×C₃⋊D₃	1	4	yes	—
275	C₃	—	2	4	no	—
276	C₃	C₂²×D₃×Dic₆	1	4	yes	complemented
277	C₃	C₂³×#634	1	1	yes	characteristic, complemented
278	C₂	—	3	8	no	—
279	C₂	C₂×D₃×#634	1	6	yes	direct product factor
280	C₂	C₂³×D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×C₃⋊D₃
commutator, cosocle	C₂×C₃³	C₂⁵
Frattini	C₂	C₂³×D₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂³×C₃³	C₂³
prime radical	C₂³×C₃²×D₃	C₂²
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₂³×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#6308	C₂	semidirect
2. C₂²×C₃×#634	C₂	semidirect
4. C₂²×#6293	C₂	semidirect
6. C₂×D₃×#634	C₂	direct (N × H)
12. C₂×#6308	C₂²	semidirect
14. C₂×C₃×#634	C₂²	semidirect
17. C₂×#6293	C₂²	semidirect
19. D₃×#634	C₂²	direct (N × H)
27. C₂²×#634	D₃	direct (N × H)
34. C₂²×C₃²×D₃	Q₈	semidirect
38. #6308	C₂³	semidirect
40. C₃×#634	C₂³	semidirect
43. #6293	C₂³	semidirect
64. C₂×#634	C₂×D₃	direct (N × H)
72. C₂²×C₃³	C₂×Q₈	semidirect
74. C₂×C₃²×D₃	C₂×Q₈	semidirect
98. C₂²×C₃×D₃	Dic₆	semidirect
114. #634	C₂²×D₃	direct (N × H)
122. C₂×C₃³	C₂²×Q₈	semidirect
124. C₃²×D₃	C₂²×Q₈	semidirect
154. C₂²×C₃²	C₂×Dic₆	semidirect
155. C₂²×C₃²	D₃×Q₈	semidirect
158. C₂×C₃×D₃	C₂×Dic₆	semidirect
176. C₃³	C₂³×Q₈	semidirect
182. C₂²×D₃	#634	direct (N × H)
208. C₂×C₃²	C₂²×Dic₆	semidirect
209. C₂×C₃²	C₂×D₃×Q₈	semidirect
212. C₃×D₃	C₂²×Dic₆	semidirect
228. C₂²×C₃	D₃×Dic₆	semidirect
229. C₂²×C₃	C₂×#634	semidirect
232. C₂×D₃	C₂×#634	direct (N × H)
245. C₃²	C₂³×Dic₆	semidirect
246. C₃²	C₂²×D₃×Q₈	semidirect
262. C₂×C₃	C₂×D₃×Dic₆	semidirect
263. C₂×C₃	C₂²×#634	semidirect
266. D₃	C₂²×#634	direct (N × H)
269. C₂²	D₃×#634	direct (N × H)
276. C₃	C₂²×D₃×Dic₆	semidirect
277. C₃	C₂³×#634	semidirect
279. C₂	C₂×D₃×#634	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#6308	2
C₂²×C₃×#634	2
C₂²×C₃²×C₄×D₃	2
C₂²×#6293	2
C₂²×D₃×C₃⋊Dic₃	2
C₂×D₃×#634	2
C₂³×#634	3
C₂²×D₃×Dic₆	3

Subgroups — C22×D3×#634 #13181589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃×#634 #13181589